Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime

Adopt the following reduced circuit model in the laboratory frame and SI units of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values. The continuum material reference is Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch; the terminal laws below are additional assumptions, rather than consequences of its pointwise constitutive relations or of Maxwell's equations.

(a) Branch data. Fix a nonempty interval $J\subseteq \mathbb{R}$, a finite set of nodes $\mathcal{N}$, and a finite set of oriented branches $\mathcal{E}$, each with specified tail and head nodes. Designate $n$ inductive/resistive branches, $m$ capacitive branches and any source branches. On the first family specify currents $I:J\to {\mathbb{R}}^n$ and separate voltage-drop vectors $V_L,V_R:J\to {\mathbb{R}}^n$; on the second specify charges $Q:J\to {\mathbb{R}}^m$, currents $I_C:J\to {\mathbb{R}}^m$ and voltage drops $V_C:J\to {\mathbb{R}}^m$. Currents $I$ and charges $Q$ admit $C^1$ extensions to an open interval containing $J$; included endpoint derivatives are their one-sided derivatives, and at a singleton the chosen extensions are part of the data supplying the jets. Voltages and source currents are continuous. The matrices $L,R\in {\mathbb{R}}^{n\times n}$ and $C\in {\mathbb{R}}^{m\times m}$ are constant, with $L^{\mathsf{T}}=L$, $C^{\mathsf{T}}=C$, $z^{\mathsf{T}}Cz>0$ for every $z\mathrm{\ne }0$, $R^{\mathsf{T}}=R$, and $y^{\mathsf{T}}Ry\ge 0$ for every $y$. These inequalities define positive definiteness and positive semidefiniteness here. Symmetry alone is imposed on a declared $L$; a smooth compact-current realization can supply its additional positivity. Geometry, orientations and source data are fixed.

(b) Element and network relations. Adopt $V_L=L\dot{I},V_R=RI,I_C=\dot{Q},V_C=C^{-1}Q\mathrm{.}$ Positive definiteness makes $C^{-1}$ well defined: $Cz=0$ implies $z^{\mathsf{T}}Cz=0$, hence $z=0$; an injective square finite matrix is invertible by Invertible matrix theorem: invertibility, full pivot rank, RREF $I$, trivial nullspace and unique solvability are equivalent. The voltage drop on an inductive/resistive branch is $(V_L+V_R{)}_{\alpha }$, with either element omitted by setting its corresponding matrix to zero; a capacitive branch has drop $(V_C{)}_{\beta }$. Thus the three voltage symbols refer to distinct element drops. In a purely inductive use, abbreviate $V_L$ by $V$. The linkage is $LI$ and the induced emf convention is $e_L:=-V_L$. The capacitor scalar law is $V_C=Q\mathrm{/}C$ when $m=1$.

For a branch $b$ let $i_b$ be its oriented current and $v_b$ its voltage drop from tail to head (a source may have a negative drop). Define the incidence matrix by $A_{ab}:={\mathbf{1}}_{\{a=\mathrm{tail}(b)\}}-{\mathbf{1}}_{\{a=\mathrm{head}(b)\}}$. Kirchhoff's current law is $Ai(t)=0$, expressing no node-charge accumulation. A closed walk is a finite sequence of traversals of branches with matching successive nodes and final node equal to initial node. If its signed traversal count is $z_b$ (forward minus backward), Kirchhoff's voltage law is ${\sum }_b{z}_b{v}_b(t)=0$. Adopt it for every closed walk. Equivalently it suffices to impose it on a chosen linearly independent spanning set (basis) of the real span of their traversal-count vectors; this is the meaning of independent loops here. These are imposed ideal network constraints, not statements about arbitrary field circulations.

The bookkeeping energies are $\frac{1}{2}{I}^{\mathsf{T}}LI$ and $\frac{1}{2}{Q}^{\mathsf{T}}{C}^{-1}Q$. If a smooth compact conserved-profile realization of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b) is specified, identification of its linkage matrix with the declared $L$ is an additional model premise, not automatic.

(c) Quasistatic regime. Specify a geometric size $\mathrm{\ell }>0$, a time scale $T>0$ and a chosen dimensionless tolerance $0<{ϵ}_{\mathrm{q}\mathrm{s}}<1$. The declaration $\mathrm{\ell }\mathrm{/}(cT)\ll 1$ means $\mathrm{\ell }\mathrm{/}(cT)\le {ϵ}_{\mathrm{q}\mathrm{s}}$ with a tolerance regarded as small in the chosen application. Negligible radiation and leakage, ideal terminals and the fixed conductor geometry are separately adopted idealizations. Neither this inequality nor these idealizations give a quantitative error bound for an arbitrary full Maxwell solution; no universal accuracy is asserted.

(d) Units and scope. $V_L,V_R,V_C,v_b$ carry volts, $I,I_C,i_b$ amperes, $Q$ coulombs, $t$ seconds, $L$ henries ($\mathrm{H}=\mathrm{V}\mathrm{s}{\mathrm{A}}^{-1}$), $C$ farads ($\mathrm{F}=\mathrm{C}{\mathrm{V}}^{-1}$), and $R$ ohms ($\mathrm{\Omega }=\mathrm{V}{\mathrm{A}}^{-1}$). The model excludes radiation, distributed-element, nonlinear, hysteretic, leakage and proximity effects; resistor loss is retained through $R$. Subluminal material speeds are required if a relativistic moving-matter interpretation is added. The model is adopted and contains no measurement or derivation from the vacuum Maxwell equations.

The scalar series RL loop consists of one inductive/resistive branch and a source: voltage balance gives $L\dot{I}+RI=V_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}$. The source-free scalar LC loop uses $R=0$, equal oriented currents $I=I_C=\dot{Q}$ from the node law, and voltage balance $L\dot{I}+Q\mathrm{/}C=0$. These identify the exact network assumptions used by the later conditional deductions. General network solvability is not asserted.

Tong §4.1.2–4.1.3, printed pp. 71–77, supplies inductive energy, the induced-emf sign and the adoption of Ohm's law; its worked dynamical example there is a sliding bar. Fitzpatrick §7.4–7.6, printed pp. 222–234, supplies the distinct inductive/resistive voltage drops, RL equation (7.37), and series LCR law (7.76), including capacitor current as the charge derivative. The finite network and matrix formulation here is explicitly adopted model content; no source is represented as proving a Maxwell-to-network error estimate.